Greedy sparsifications of sums of positive semidefinite matrices

TL;DR

This paper establishes a deterministic method to approximate the identity matrix as an average of selected positive semidefinite matrices, improving upon prior randomized sampling results with explicit guarantees.

Contribution

It provides a deterministic analogue to Rudelson's sampling theorem for sums of positive semidefinite matrices, with explicit bounds on approximation quality.

Findings

Existence of a deterministic sequence approximating the identity matrix within specified bounds.

Explicit bounds depend on the number of matrices and their spectral norm.

The method guarantees an approximation within any desired error with a sufficient number of matrices.

Abstract

We prove a deterministic analogue of Rudelson's sampling theorem for sums of positive semidefinite matrices. Let $A_1,\dots,A_m$ be positive semidefinite \(d\times d\) matrices, and let $\lambda_1,\dots,\lambda_m \ge 0$ satisfy \[ \sum_{i=1}^m \lambda_i = 1, \qquad \sum_{i=1}^m \lambda_i A_i = I_d, \qquad \|A_i\| \le M \quad\text{for all } i=1,\dots,m. \] We show that there exists a deterministic sequence of indices $i_1,i_2,\dots \in \{1,\dots,m\}$ such that for every integer $k \ge 1$, \[ \left\| \frac{1}{k}\sum_{r=1}^k A_{i_r} - I_d \right\| \le \begin{cases} \displaystyle \frac{2M\ln(2d)}{k}, & \text{if } k \le M\ln(2d),\\[2ex] \displaystyle 3\sqrt{\frac{M\ln(2d)}{k}}, & \text{if } k > M\ln(2d). \end{cases} \] In particular, if $0<\varepsilon\le 1$ and $N \ge 9M\ln(2d)\varepsilon^{-2}$, then one can choose indices $i_1,\dots,i_N \in \{1,\dots,m\}$ such that \[ \left\| \frac{1}{N}\sum_{r=1}^N A_{i_r} - I_d \right\| \le \varepsilon. \]